Problems In Philosophy • 4

Posted on February 9, 2016 by Jon Awbrey

Re: R.J. Lipton and K.W. Regan • Did Euclid Really Mean ‘Random’?

These are the forms of time,
which imitates eternity and
revolves according to a law
of number.

Plato • Timaeus • 38 A
Benjamin Jowett (trans.)

It is clear from Aristotle and places in Plato that the good of reasoning from fair samples and freely chosen examples was bound up with notions of probability, which in the Greek idiom meant likeness, likelihood, and likely stories, the question being how much the passing image could tell us of the original idea.

Posted in Aristotle, Computation, Computer Science, Euclid, Genericity, Geometry, Iconicity, Likelihood, Likely Story, Likeness, Mathematics, Number Theory, Philosophy, Philosophy of Mathematics, Plato, Probability, Socrates | Tagged , , , , , , , , , , , , , , , , | 3 Comments

Problems In Philosophy • 3

Posted on February 8, 2016 by Jon Awbrey

Re: R.D. Mounce

Making reality our friend is necessary to survival and finding good descriptions of reality is the better part of doing that, so I don’t imagine we have any less interest in truth than the Ancients.  From what I remember of him, Plato had specific objections to specific styles of art, poetry, or their interpretation, and hardly painted art in general with a broad brush.  Indeed, there is a Pythagorean tradition that reads The Republic as a metaphorical treatise on musical theory, applied no doubt to achieving harmony in society.  Truth in fiction and myth is a matter of interpretation, and come to think of it, that’s not essentially different from truth in more literal forms of expression.

Posted in Aesthetics, Computation, Computer Science, Ethics, Heap Problem, Logic, Mathematics, Model Theory, Normative Science, Paradox, Peirce, Philosophy, Pragmatism, Problem Solving, Proof Theory, Sorites | Tagged , , , , , , , , , , , , , , , | Leave a comment

Problems In Philosophy • 2

Posted on February 8, 2016 by Jon Awbrey

Re: R.J. Lipton and K.W. Regan • You Think We Have Problems

Classical tradition views logic as a normative science, one whose object is truth.  This puts logic on a par with ethics, whose object is justice or morality in action, and aesthetics, whose object is beauty or the admirable for its own sake.

The pragmatic spin on this line of thinking treats logic, ethics, aesthetics as a concentric series of normative sciences, each a subdiscipline of the next.  Logic tells us how we ought to conduct our reasoning in order to achieve the goals of reasoning in general.  Thus logic is a special case of ethics.  Ethics tells us how we ought to conduct our activities in general in order to achieve the good appropriate to each enterprise.  What makes the difference between a normative science and a prescriptive dogma is whether this telling is based on actual inquiry into the relationship of conduct to result, or not.

Here’s a bit I wrote on this a long time ago in a galaxy not far away —

Posted in Aesthetics, Algorithms, Animata, Automata, Beauty, C.S. Peirce, Ethics, Inquiry, Justice, Logic, Model Theory, Normative Science, Peirce, Philosophy, Pragmatism, Problem Solving, Proof Theory, Summum Bonum, Truth, Virtue | Tagged , , , , , , , , , , , , , , , , , , , | 3 Comments

Problems In Philosophy • 1

Posted on February 7, 2016 by Jon Awbrey

Re: R.J. Lipton and K.W. Regan • You Think We Have Problems

I used to think about the heap problem a lot when I was programming and I decided the heap quits being a heap as soon as you remove one grain because then it becomes two heaps.

The Pascal sorting of the sorites played on moves between heaps and stacks, but I’ve forgotten the details of that epiphany.  The whole-systems point is clear enough though — the system as a whole makes a discrete transition from one state of organization to another.

Posted in Computation, Computer Science, Heap Problem, Paradox, Philosophy, Sorites | Tagged , , , , , | 1 Comment

Riffs and Rotes • 3

Posted on February 3, 2016 by Jon Awbrey

Re: R.J. Lipton • Failure Of Unique Factorization

My favorite question in this realm is how much of the linear ordering of the natural numbers is purely combinatorial, where we eliminate all the structure that isn’t purely combinatorial via the doubly recursive factorizations of whole numbers, ending up with two species of graph-theoretic structures that I dubbed Riffs and Rotes.  See the following links for more discussion:

Posted in Algebra, Combinatorics, Graph Theory, Group Theory, Logic, Mathematics, Number Theory, Riffs and Rotes | Tagged , , , , , , , | Leave a comment

Supple Agony

Posted on January 25, 2016 by Jon Awbrey

A lack of knowing, my lack of knowing
How to supply either lack of knowing.
Complemental and supplemental ∠s.

Posted in Anamnesis, Communication, Learning, Maieusis, Mantra, Meditation, Memo, Meno, Teaching | Tagged , , , , , , , , | Leave a comment

Prospects for Inquiry Driven Systems • 1

Posted on January 22, 2016 by Jon Awbrey

I finally finished retyping the bibliography to my systems engineering proposal that had gotten lost in a move between computers, so here is a link to the OEIS Wiki copy.

This may be of interest to people working towards applications of Peirce’s theory of inquiry, especially the design of intelligent systems with a capacity for supporting scientific inquiry.

Posted in Adaptive Systems, Animata, Artificial Intelligence, Automated Research Tools, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Cybernetics, Differential Logic, Educational Systems Design, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Intelligent Systems Engineering, Learning, Logic, Logic of Science, Logical Graphs, Machine Learning, Peirce, Propositional Calculus, Reasoning, Scientific Method, Semiotics, Sign Relations, Theorem Proving | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , | 2 Comments

Relations & Their Relatives • Discussion 17

Posted on January 4, 2016 by Jon Awbrey

Re: Peirce List Discussion • HR

We have been considering special properties that a dyadic relation may have, in particular, the following two symmetry properties.

  • A dyadic relation L is symmetric if (x, y) being in L implies that (y, x) is in L.
  • A dyadic relation L is asymmetric if (x, y) being in L implies that (y, x) is not in L.

The first thing to understand about any symmetry of any relation is that it is a property of the whole relation, the whole set of tuples, not a property of individual tuples.

Many properties of dyadic relations can be made visually evident by arranging their ordered pairs in 2-dimensional arrays.  Let’s do this for our initial sample of biblical brothers, using the first three letters of their names as row and column labels.

The relation B indicated by “brother of” is a symmetric relation.  The ordered pairs of B are given below.

\begin{array}{l|*{8}{c}} B & \rm{Abe} & \rm{Ben} & \rm{Cai} & \rm{Esa} & \rm{Isa} & \rm{Ish} & \rm{Jac} & \rm{Jos} \\[2pt] \hline \\[2pt] \rm{Abe} & \centerdot && \rm{Abe:Cai} &&&&&\\[12pt] \rm{Ben} && \centerdot &&&&&& \rm{Ben:Jos} \\[12pt] \rm{Cai} & \rm{Cai:Abe} && \centerdot &&&&&\\[12pt] \rm{Esa} &&&& \centerdot &&& \rm{Esa:Jac} &\\[12pt] \rm{Isa} &&&&& \centerdot & \rm{Isa:Ish} &&\\[12pt] \rm{Ish} &&&&& \rm{Ish:Isa} & \centerdot &&\\[12pt] \rm{Jac} &&&& \rm{Jac:Esa} &&& \centerdot &\\[12pt] \rm{Jos} && \rm{Jos:Ben} &&&&&& \centerdot \end{array}

The relation E indicated by “elder brother of” is an asymmetric relation.  The ordered pairs of E are given below.

\begin{array}{l|*{8}{c}} E & \rm{Abe} & \rm{Ben} & \rm{Cai} & \rm{Esa} & \rm{Isa} & \rm{Ish} & \rm{Jac} & \rm{Jos} \\[2pt] \hline \\[2pt] \rm{Abe} & \centerdot &&&&&&&\\[12pt] \rm{Ben} && \centerdot &&&&&&\\[12pt] \rm{Cai} & \rm{Cai:Abe} && \centerdot &&&&&\\[12pt] \rm{Esa} &&&& \centerdot &&& \rm{Esa:Jac} &\\[12pt] \rm{Isa} &&&&& \centerdot &&&\\[12pt] \rm{Ish} &&&&& \rm{Ish:Isa} & \centerdot &&\\[12pt] \rm{Jac} &&&&&&& \centerdot &\\[12pt] \rm{Jos} && \rm{Jos:Ben} &&&&&& \centerdot \end{array}

Posted in C.S. Peirce, Dyadic Relations, Logic, Logic of Relatives, Mathematics, Peirce, Peirce List, Relation Theory, Semiotics, Sign Relations, Triadic Relations | Tagged , , , , , , , , , , | 12 Comments

Looking Back On 2015

Posted on December 31, 2015 by Jon Awbrey

The WordPress.com stats helper monkeys prepared a 2015 annual report for this blog.

Here's an excerpt:

The concert hall at the Sydney Opera House holds 2,700 people. This blog was viewed about 31,000 times in 2015. If it were a concert at Sydney Opera House, it would take about 11 sold-out performances for that many people to see it.

Click here to see the complete report.

Posted in Annual Report, Year In Review | Tagged , | Leave a comment

Considerate Reason • 2

Posted on December 29, 2015 by Jon Awbrey

Re: R.J. Lipton • Why Is Discrete Math Hard To Teach?

The Liberal Arts trivium of Grammar, Logic, Rhetoric received a latter day echo in the Unified Science trivium of Syntax, Semantics, Pragmatics, which was in turn the way Charles Morris transmogrified C.S. Peirce’s theory of triadic sign relations.

Resources

Posted in Argument, C.S. Peirce, Computer Programming, Discrete Mathematics, Education, Educational Systems Design, Grammar, Inquiry, Inquiry Driven Systems, Interpretation, Logic, Logic of Relatives, Mathematics, Peirce, Pragmatics, Relation Theory, Rhetoric, Semantics, Semiotics, Sign Relations, Syntax, Triadic Relations | Tagged , , , , , , , , , , , , , , , , , , , , , | Leave a comment